Eigen-oscillations of contrasting non-homogeneous elastic bodies: asymptotic and uniform estimates for eigenvalues

Estimates of convergence rates for the eigenvalues of spectralstiff elasticity problems are obtained. The bounds in the estimatesare expressed in terms of the stiffness ratio h and characteristicproperties of the limit spectrum for low and middle frequencyranges. These estimates allow us to distinguish between individualand collective asymptotics of the eigenvalues and eigenvectorsand to determine precisely the intervals for the small parameterh where the mathematical model considered provides a suitableapproach and accuracy. The results in this paper hold for differentboundary conditions, two- and three-dimensional models and scalarproblems.     

Locking-free DGFEM for elasticity problems in polygons

The h-version of the discontinuous Galerkin finite element method(h-DGFEM) for nearly incompressible linear elasticity problemsin polygons is analysed. It is proved that the scheme is robust(locking-free) with respect to volume locking, even in the absenceof H²-regularity of the solution. Furthermore, it is shown thatan appropriate choice of the finite element meshes leads torobust and optimal algebraic convergence rates of the DGFEMeven if the exact solutions do not belong to H².     

Eigenvalues of periodic Sturm-Liouville problems by the Shannon-Whittaker sampling theorem

We are concerned with the computation of eigenvalues of a periodic Sturm-Liouville problem using interpolation techniques in Paley-Wiener spaces. We shall approximate the Hill discriminant by sampling a few of its values and then find its zeroes which are the square roots of the eigenvalues. Computable error estimates are provided together with eigenvalue enclosures.

     

Oscillation theory and numerical solution of fourth-order Sturm--Liouville problems

A shooting method is developed to approximate the eigenvaluesand eigenfunc-tions of a fourth-order Sturm-Liouville problem.The main tool is a miss-distance function M(), which countsthe number of eigenvalues less than A. The method approximatesthe coefficients of the differential equation by piecewise-constantfunctions, which enables an exact solution to be found on eachmesh interval. In order to calculate N() for the approximateproblem, certain oscillation numbers N_L and N_R must be computed.These consist of sums of nullities (or rank deficiencies) of2 x 2 matrices obtained from the solutions of the approximatedifferential equation. Although these solutions can be foundexplicitly, the calculation of N_L and N_R is non-trivial, andis obtained by using certain properties of M().     

Use of extrapolation for improving the order of convergence of eigenelement approximations

We consider the approximation of the eigenelements of a compactintegral operator defined on C[0, 1] with a smooth kernel. Weuse the iterated collocation method based on r Gauss pointsand piecewise polynomials of degree r – 1 on each subintervalof a nonuniform partition of [0, 1]. We obtain asymptotic expansionsfor the arithmetic means of m eigenvalues and also for the associatedspectral projections. Using Richardson extrapolation, we showthat the order of convergence O(h^2r) in the iterated collocationmethod can be improved to O(h^2r+2). Similar results hold forthe Nyström method and for the iterated Galerkin method.We illustrate the improvement in the order of convergence bynumerical experiments.     

Eigenvalues of s-l systems using sampling theory

A new method based on Shannon's sampling theorem is introduced for the localization and approximation of the eigenvalues of regular Sturm-Liouville problems. Several examples have been presented to illustrate the effectiveness of the method.     

A new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-valueproblems for linear evolution partial differential equationswith spatial derivatives of arbitrary order. This method isillustrated by solving several such problems on the half-line{t > 0, 0 < x < }, and on the quarter-plane {t >0, 0 < x_j < , j = 1, 2}. For equations in one space dimensionthis method constructs q(x, t) as an integral in the complexk-plane involving an x-transform of the initial condition anda t-transform of the boundary conditions. For equations in twospace dimensions it constructs q(x₁, x₂, t) as an integral inthe complex (k₁, k₂)-planes involving an (x₁, x₂)-transformof the initial condition, an (x₂, t)-transform of the boundaryconditions at x₁ = 0, and an (x₁, t)-transform of the boundaryconditions at x₂ = 0. This method is simple to implement andyet it yields integral representations which are particularlyconvenient for computing the long time asymptotics of the solution.     

A New Fourth-order Finite-difference Method for Computing Eigenvalues of Fourth-order Two-point Boundary-value Problems

We present a new finite-difference method for computing eigenvaluesof two-point boundary-value problems involving a fourth-orderdifferential equation. Our finite-difference method leads toa generalized seven-diagonal symmetric-matrix eigenvalue problemand provides O(h⁴)-convergent approximations for the eigenvalues.     

A Minimax Principle for the Eigenvalues in Spectral Gaps

A minimax principle is derived for the eigenvalues in the spectralgap of a possibly non-semibounded self-adjoint operator. Itallows the nth eigenvalue of the Dirac operator with Coulombpotential from below to be bound by the nth eigenvalue of asemibounded Hamiltonian which is of interest in the contextof stability of matter. As a second application it is shownthat the Dirac operator with suitable non-positive potentialhas at least as many discrete eigenvalues as the Schrödingeroperator with the same potential.     

A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix

A fast method for computing all the eigenvalues of a Hamiltonian matrix M is given. The method relies on orthogonal symplectic similarity transformations which preserve structure and have desirable numerical properties. The algorithm requires about one-fourth the number of floating-point operations and one-half the space of the standard QR algorithm. The computed eigenvalues are shown to be the exact eigenvalues of a matrix M + E where ∥E∥ depends on the square root of the machine precision. The accuracy of a computed eigenvalue depends on both its condition and its magnitude, larger eigenvalues typically being more accurate.     

Higher Approximation of Eigenvalues by the Sampling Method

We study the rate of convergence of the sampling method, which deals with the computation of eigenvalues of regular Sturm-Liouville problems. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Numerical examples are also provided.     

Solution of Non-linear Eigenvalue Problems by the Continuation Method

The Problem of finding the roots (eigenvalues) of the equationdet A()=0, where A in an nxn matrix, is studied. There existseveral efficient local iterative methods for this problem.However, no efficient global method is available. We describethe application of the continuation method to this problem andsolve two examples by it. We conclude that the continuationmethod is a practical global strategy for locating eigenvaluesof non-linear matrices. This method is even more effective whenit is combined with an appropriate iterative scheme.     

Parameter-free H(div) preconditioning for a mixed finite element formulation of diffusion problems

^** Email: c.powell{at}manchester.ac.uk Mixed finite element formulations of generalised diffusion problemsyield linear systems with ill-conditioned, symmetric and indefinitecoefficient matrices. Preconditioners with optimal work complexitythat do not rely on artificial parameters are essential. Weimplement lowest order Raviart–Thomas elements and analysepractical issues associated with so-called ‘H(div) preconditioning’.Properties of the exact scheme are discussed in Powell &Silvester (2003, SIAM J. Matrix Anal. Appl., 25, 718–738).We extend the discussion, here, to practical implementation,the components of which are any available multilevel solverfor a weighted H(div) operator and a pressure mass matrix. Anew bound is established for the eigenvalue spectrum of thepreconditioned system matrix and extensive numerical resultsare presented.     

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C² solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.     

Multiplicative Error Analysis of Matrix Transformation Algorithms

The author's recently introduced relative error measure forvectors is applied to the error analysis of algorithms whichproceed by successive transformation of a matrix. Instead ofmodelling the roundoff errors at each stage by A: = T(A)+E onemodels them by A: =e^E T(A) where E is a small linear transformation.This can simplify analyses considerably. Applications to theparallel Jacobi method for eigenvalues, and to Gaussian elimination,are given.     

On Hill's Equation with a Singular Complex-Valued Potential

In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x₀, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C². We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.     

The Rotation Number Approach to Eigenvalues of the One-Dimensional p-Laplacian with Periodic Potentials

The paper studies the periodic and anti-periodic eigenvaluesof the one-dimensional p-Laplacian with a periodic potential.After a rotation number function () has been introduced, itis proved that for any non-negative integer n, the endpointsof the interval ^–1(n/2) in R yield the corresponding periodicor anti-periodic eigenvalues. However, as in the Dirichlet problemof the higher dimensional p-Laplacian, it remains open if theseeigenvalues represent all periodic and anti-periodic eigenvalues.The result obtained is a partial generalization of the spectrumtheory of the one-dimensional Schrödinger operators withperiodic potentials.     

The inverse eigenvalue problem for a weighted helmholtz equation

Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ^* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

Computing Derivatives of Eigensystems by the Vector {varepsilon}-Algorithm

Rudisill & Chu proposed a (slowly converging) iterativemethod for computing partial derivatives of eigenvalues andeigenvectors of parameter-dependent matrices. It is shown that,with exact computation, application of the vector -algorithmto this method produces the exact solution in a small numberof steps. Numerical results demonstrate the viability of thismethod. A refinement process is suggested which makes the methodespecially effective for subdominant eigenvalues.